Tensor diagrams and cluster algebras

Sergey Fomin, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G. Kuperberg.

Original languageEnglish (US)
Pages (from-to)717-787
Number of pages71
JournalAdvances in Mathematics
StatePublished - Sep 10 2016

Bibliographical note

Funding Information:
Partially supported by NSF grants DMS-1101152 , DMS-1361789 (S.F.) and DMS-1068169 , DMS-1351590 (P.P.).

Publisher Copyright:
© 2016 Elsevier Inc.

Copyright 2016 Elsevier B.V., All rights reserved.


  • Cluster algebra
  • Invariant theory
  • Tensor diagram
  • Web basis


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